Weird $\mathbb{R}$-Factorizable Groups

Abstract

The problem of the existence of non-pseudo-${\aleph }_1$-compact $\mathbb{R}$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $\ddot{I}{‰}_1$. Closely related results concerning the $\mathbb{R}$-factorizability of products of topological groups and spaces are also obtained (a product $X\times Y$ of topological spaces is said to be $\mathbb{R}$-factorizable if any continuous function $X\times Y\to \mathbb{R}$ factors through a product of maps from $X$ and $Y$ to second-countable spaces). In particular, it is proved that the square $G\times G$ of a topological groups $G$ is $\mathbb{R}$-factorizable as a group if and only if it is $\mathbb{R}$-factorizable as a product of spaces, in which case $G$ is pseudo-${\aleph }_1$-compact. It is also proved that if the product of a space $X$ and an uncountable discrete space is $\mathbb{R}$-factorizable, then $X^{\ddot{I}}‰$ is heredirarily separable and heredirarily Lindelöf.